[jasomill]

Does anyone know (or can refer to) a reasonable definition of +,-,*,/,% (in their usual meanings) on rational numbers such that A op B always evaluates to a rational?

Given the usual definition of multiplication (and division as its inverse), this is not possible:

Assume some multiplicative inverse 0⁻¹ of 0 exists. Then, writing · for multiplication, since

a·0 = b·0

for all rational a and b, we have

a·0·0⁻¹ = b·0·0⁻¹

and therefore

a = b

for all rational a and b, which is absurd.

[jasomill]

As for extending the rationals by 0⁻¹ and giving up properties, a·0 = 0 can still hold for all a ≠ 0⁻¹.

To get an idea of what operations do and don't make sense when extending the rationals in the way you propose, define ∞ as our 0⁻¹ and refer to

https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

substituting ℚ and ℚ̂ for ℝ and ℝ̂.

For a similar construction that maintains ordering properties, see

https://en.wikipedia.org/wiki/Extended_real_number_line#Arit...

Finally, note that the complex number equivalent to your construction is really interesting and useful:

https://en.wikipedia.org/wiki/Riemann_sphere

https://www-users.cse.umn.edu/~arnold/moebius/

[JonChesterfield]

The Riemann sphere has always existed beyond my grasp of mathematics. I remember the maths undergraduates being excited about it. Thank you for the references, it looks like I need to learn more before proceeding with certainty.